Proposition. Pseudomonoids in span(set) are precisely discrete promonoidal categories [mde-Z2KV]
Proposition. Pseudomonoids in span(set) are precisely discrete promonoidal categories [mde-Z2KV]
A promonoidal structure on a discrete category \(\mathbb {C}\) amounts to families of sets \(\{\mathbb {C}(A,B;C)\}_{A,B,C \in {\mathbb {C}}}\) and \(\{\mathbb {C}(;A)\}_{A \in {\mathbb {C}}}\) along with coherent associator and unitor isomorphisms. We can repackage this data as spans of sets \(\mathbb {C} \leftarrow \sum _{A,B,C} \mathbb {C}(A,B;C) \to \mathbb {C}\) and \(1 \leftarrow \sum _{A} \mathbb {C}(;A) \to \mathbb {C}\), and the associator and unitor isomorphisms express that these spans \(\mathbb {C} \times \mathbb {C} \to \mathbb {C}\), \(1 \to \mathbb {C}\) form a pseudomonoid.